In Exercises 1–26, find the exact value of each expression. _ ...

Question

In Exercises 1–26, find the exact value of each expression. _ csc⁻¹ (− 2√3/3)

Accepted Answer

Hello everybody. I hope you're doing. All right. Today, today we're going to be looking at this question that states for the following trigonometric expression evaluate and give the answer as an exact value in radiance where the trigonometric expression states inverse cos of negative square root of two. Now the answer choices provided are a negative pi divided by two B negative pi divided by three C negative pi divided by four and D negative pi divided by six. Now for a question like this, let's set up an if and then situation. So let's see if inverse sea cans of negative square root of two is equal to X. Then regular cos cans of X is equal to the negative squared of two. Now what is co C can? Well cos cans of is equal to one divided by sine of X. So it's the reciprocal of sine. Therefore sin of X in this case will be equal to negative one divided by square root of two, which can be rewritten as negative squared of two divided by two. Now, with that information, we can turn to the unit circle and we can recall the different values that we know using it to our advantage. So we calling from the unit circle, let's say we have pi divided by four. Now pi divided by four, that angle has a point at positive squared with an X component at positive squared of two divided by two comma positive squared of two divided by two as the Y component as well. Now something we should note is that sign of theta is equal to or is associated with the Y values. Therefore, if we have, let's say sign of pi divided by four, that will be equal to the Y component at pi divided by four, which we just said was squared of two divided by two. Now we need the negative version of this because we want sign of X equals negative squared of two divided by two and we have positive squared of two divided by two. So we, we multiply each side by a negative and we have negative sign of pi divided by four equals negative squared of two divided by two. Now, we can not that negative sign of theta is equal to a sign of negative theta and we can rewrite what we have. So we have negative sign of pi divided by four, that'll be equal to sign of negative pi divided by four, which is equal to negative squared of two divided by two as we said. Therefore, we had NS sign of X equals negative squared of two divided by two And now we have sine of negative pi divided by four equals negative squared of two divided by two. Therefore, we have that the X in our problem is equal to negative pi divided by four. And just like that, we have been able to solve our problem. So we can see how important it is to be able to recall the different values from the unit circle as well as noting different ways that we can rewrite our equations. So I hope that was helpful. And until next time.

In Exercises 1–26, find the exact value of each expression. _ csc⁻¹ (− 2√3/3)

Key Concepts

Inverse Cosecant Function (csc⁻¹)

Relationship Between Cosecant and Sine

Exact Values of Special Angles

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